Energy spectrum of confined positively charged excitons in single quantum dots

A theoretical model which relates the binding energy of a positively charged exciton in a quantum dot with the confinement energy is presented. It is shown that the binding energy, defined as the energy difference between the corresponding charged and neutral complexes confined on the same excitonic shell, strongly depends on the shell index. Moreover, we found that the ratio of the binding energy for positively charged excitons from the $p$ and $s$ shells of a dot depends mainly on the nearly perfect confinement inside the dot, which is due to the “hidden symmetry” of the multielectron-hole system. We applied the theory to the excitons confined to a single GaAlAs/AlAs quantum dot. The relevant binding energy was determined using microphotoluminescence and microphotoluminescence excitation magnetospectroscopy. We show that within our theory, the confinement energy determined using the ratio of the binding energy corresponds well to the actual confinement energy of the investigated dot.

Figure 1

Absorption spectra within the $s,p$, and $d$ shells of a single QD parametrized by ${\omega }_e=20$ meV and ${\omega }_h=4$ meV in the absence of a magnetic field when (a) all intershell Coulomb scattering matrix elements are ignored and all Coulomb matrix elements within (b) two $\left(s,p\right)$, (c) three $\left(s,p,d\right)$, and (d) eight $(s,p,d$, …) electron and hole shells are included. Note that (b) and (c) display absorption spectra limited to the lowest two and three shells, respectively, because all higher shells were excluded from the calculation. Gray and blue (red) peaks represent the resonances related to the neutral exciton $\left(1e_{x}1h_x\right)$ and the singlet- (triplet-) spin state of the positively charged exciton $\left(1e_{x}1h_{x}1h_s\right)$, respectively. The horizontal blue and red arrows indicate the magnitude of binding energy at the $s,p$, and $d$ shells $({\mathrm{\Delta }}_{ss},{\mathrm{\Delta }}_{pp}$, and ${\mathrm{\Delta }}_{dd}$, respectively).

Figure 2

Color-coded contour plot map of the binding energy ratio $R={\mathrm{\Delta }}_{pp}/{\mathrm{\Delta }}_{ss}$ in coordinates ${\omega }_e+{\omega }_h$ and ${\omega }_e/{\omega }_h$.

Figure 3

The PL spectra of a single GaAlAs QD recorded for two perpendicular linear polarizations oriented along the crystallographic directions $\left[110\right]$ (orange curve) and $\left[1\overline{1}0\right]$ (green curve).

Figure 4

The PLE spectra detected on the $\mathrm{X}$ (green curve) and ${\mathrm{X}}^{+}$ (orange curve) emission lines of a single GaAlAs QD. The spectra are normalized to the most intense peaks and shifted for clarity.

Figure 5

The magnetic-field evolution of PLE spectra detected on the (a) $\mathrm{X}$ and (b) ${\mathrm{X}}^{+}$ emission lines. The red and blue curves indicate the ${\sigma }_{+}$- and ${\sigma }_{-}$-polarized components of the lines, on which the PLE spectra were measured. The scale of the vertical axis is set by the energy relative to the X emission line at 0 T. The spectra are normalized to the most intense peaks and shifted for clarity. Gray lines are guides to the eye.